65.6.4 problem 4

Internal problem ID [15697]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.2, page 63
Problem number : 4
Date solved : Thursday, October 02, 2025 at 10:23:17 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=x \,{\mathrm e}^{y-x^{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 19

ode:=diff(y(x),x) = x*exp(y(x)-x^2); 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \ln \left (2\right )-\ln \left (1+{\mathrm e}^{x^{2}}\right )+x^{2} \]

✓ Mathematica. Time used: 1.914 (sec). Leaf size: 21

ode=D[y[x],x]==x*Exp[y[x]-x^2]; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\log \left (\frac {1}{2} \left (e^{-x^2}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(-x**2 + y(x)) + Derivative(y(x), x),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \log {\left (- \frac {e^{x^{2}}}{- e^{x^{2}} - 1} \right )} + \log {\left (2 \right )} \]